Method for the higher-order blind demodulation of a linear waveform transmitter

ABSTRACT

Process for the blind demodulation of a linear waveform source or transmitter in a system comprising one or more sources and an array of sensors and a propagation channel. The process comprises at least the following steps:
         the symbol period T is determined and sampled to Te such that T=IT e  (I being an integer);   a spatio-temporal observation z(t), the mixed sources of which are symbol trains from the transmitter, is constructed from the observations x(kT e );   an ICA-type method is applied to the observation vector z(t) in order to estimate the L c  symbol trains {a m-i } that are associated with the channel vectors ĥ z,j =ĥ z (k j );   the L c  outputs (â m,j , ĥ z,j ) are arranged in the same order as the inputs (a m-i ,h z (i)) so as to obtain the propagation channel vectors ĥ z,j =ĥ z (k j ); and   the phase α imax  associated with the outputs is determined.

The object of the invention relates to a process for the blinddemodulation of signals output by several transmitters and received byan array made up of at least one sensor.

For example, it applies to an array of antennas in an electromagneticcontext.

The subject of the invention is in particular the demodulation ofsignals, that is to say the extraction of the symbols {a_(k)}transmitted by a linearly modulated transmitter.

FIG. 1 shows an antenna processing system comprising severaltransmitters E_(i) and an antenna processing system T comprising severalantennas R_(i) receiving from radio sources at different angles ofincidence. The angles of incidence of the sources or transmitters may beparameterized either in 1D with the azimuth θ_(m), or in 2D with theazimuth angle θ_(m) and the elevation angle Δ_(m).

FIG. 3 shows schematically a modulation/demodulation principle for thesymbols {a_(k)} output by a transmitter. The signal propagates via amultipath channel. The transmitter outputs the symbol a_(k) at theinstant k.T, where T is the symbol period. The demodulation consists inestimating and in detecting the symbols in order to obtain, at theoutput of the demodulator, the estimated symbols â_(k). In this figure,the train of symbols {a_(k)} is linearly filtered upon transmission by atransmission filter H, also called a wave-shaping filter h₀(t).

In the rest of the description, the expression “blind demodulation” isunderstood to mean techniques that basically use no information on thesignal output, examples being a wave-shaping filter, a learningsequence, etc.

The last ten years have seen the development of SIMO (single-input,multiple-output) blind demodulation techniques called subspacetechniques using 2nd-order statistics, as described in reference [7].However, these algorithms have the drawback of not being robust toeither underestimation or overestimation of the order of the propagationchannel, resulting in temporal spreading dependent on the multipaths andon the wave-shaping filter. A linear prediction technique, described inreference [11], has been proposed for overcoming this problem, but thishas the drawback of being less effective when the length of the channelis known. To improve the subspace techniques, the method described in[16] proposes a parametric technique, but unfortunately this requiresknowledge of the wave-shaping filter.

In reference [13], the authors propose a technique based on covariancematching, but this has in particular the drawback of being verydifficult to implement. An easier but suboptimal technique, described inreference [12], was therefore developed by minimizing a likelihoodcriterion and assuming the symbols to be Gaussian in character. Thisassumption is not verified for the widely used linear modulations suchas PSK (Phase Shift Keying) or QAM (Quadrature Amplitude Modulation).

It is also known, in CMA (Constant Modulus Algorithm) methods, to use aspatio-temporal approach described for example in reference [6].However, this family of methods has the drawback of being suitable onlyfor one particular class of modulations, such as PSK, which areconstant-modulus modulations. This method is iterative and therefore hasthe drawback of having to be correctly initialized. Finally, the CMAmethods have the disadvantage of converging more slowly than theabovementioned subspace method. Moreover, reference [20] describes asubspace method making use of higher-order statistics fornon-minimum-phase FIR (finite impulse response) channel identification.

The subject of the present invention is a process based in particular onblind source separation techniques known to those skilled in the art anddescribed for example in references [4], [5], [15] and [19] assumingthat the symbols transmitted are statistically independent. To do this,the process constructs a spatio-temporal observation whose mixed sourcesare symbol trains from the transmitter. Each symbol train is for examplethe same symbol train but shifted by an integral number of symbolperiods T.

The invention relates to a process for the blind demodulation of alinear-waveform source or transmitter in a system comprising one or moresources and an array of sensors and a propagation channel, said processbeing characterized in that it comprises at least the following steps:

-   -   the symbol period T is determined and samples are taken at T_(e)        such that T=IT_(e) (I being an integer);    -   a spatio-temporal observation z(t), the mixed sources of which        are symbol trains from the transmitter, is constructed from the        observations x(kT_(e));    -   an ICA-type method is applied to the observation vector z(t) in        order to estimate the L_(c) symbol trains {a_(m-i)} that are        associated with the channel vectors ĥ_(z,j)=ĥ_(z)(k_(j));    -   the L_(c) outputs (â_(m,j), ĥ_(z,j)) are arranged in the same        order as the inputs (a_(m-i), h_(z)(i)) so as to obtain the        propagation channel vectors ĥ_(z,j)=ĥ_(z)(k_(j)); and    -   the phase α_(imax) associated with the outputs is determined.

The process according to the invention offers in particular thefollowing advantages:

-   -   it makes no assumption about the symbol constellations, unlike        the methods described in the prior art;    -   it requires no knowledge of the wave-shaping filter;    -   the modulus of the symbols is not assumed to be constant;    -   it is robust to channel length overestimation;    -   it can handle propagation channels with correlated paths; and    -   it is direct and simple to implement, with no correlated-path        crosscheck step.

Other features and advantages of the subject of the present inventionwill become more clearly apparent on reading the following description,given by way of illustration but implying no limitation, and onexamining the appended figures, which show:

FIG. 1, an example of an architecture;

FIG. 2, the angles of incidence of the sources;

FIG. 3, the linear modulation and demodulation process for a symboltrain;

FIG. 4, the diagram of a linear-modulation transmitter;

FIG. 5, a summary of the general principle employed in the invention;

FIG. 6, the representation of a constellation;

FIG. 7, a first example of the implementation of the method, in whichthe signal is received baseband;

FIG. 8, a second example in which the signal is received in baseband andthe multipaths are decorrelated; and

FIG. 9, a third example in which the signal is received in baseband andthe multipaths are groupwise decorrelated.

To explain the process according to the invention more clearly, thefollowing description relates to a process for the higher-order blinddemodulation of a linear-waveform transmitter in an array having astructure as described in FIG. 1 for example.

Before explaining the steps for implementing the process, the model ofthe signal used will be described.

Model of the Signal Output by a Source or Transmitter Linear Modulation

FIGS. 3 and 4 show the process for the linear modulation of a symboltrain {a_(k)} at the rate T by a wave-shaping filter h₀(t).

The comb of symbols c(t) is firstly filtered by the wave-shaping filterh₀(t) and then transposed to the carrier frequency f₀. The NRZ filter,which is a time window of length T, very often defined byh₀(t)=Π_(T)(t−T/2), is one particular nonlimiting example of atransmission filter. In radio communication, it is also possible to usea Nyquist filter, the Fourier transform of which, h₀(f)≈Π_(B)(f−B/2),approaches a band window B when the roll-off is zero, thereforeh₀(f)=Π_(B)(f−B/2) (the roll-off defines the slope of the filter awayfrom the band B).

The modulates signal s₀(t), output by the transmitter, may be expressedat time t_(k)=kT_(e) (T_(e) being the sampling period) as a function ofthe comb of symbols c(t):

$\begin{matrix}{{s_{0}( {kT}_{e} )} = {\sum\limits_{i}^{\;}{{h_{0}( {iT}_{e} )}{c( {( {k - i} )T_{e}} )}}}} & (1)\end{matrix}$

Let the symbol time T be equal to an integer number of times thesampling period, i.e. T=IT_(e), and let k=mI+j where 0≦j<I. Sincec(t)=Σ_(r)a_(r)δ(t−rIT_(e)), in other words c(t)=a_(u) for t=uIT_(e) andc(t)=0 for t≠uIT_(e), the only values for i for which c((k−i)T_(e)) isnon zero satisfy the equation k−i=uI, that is to say such thati=mI+j−ul=nI+j, where n=m−u. Finally, equation (1) becomes:

$\begin{matrix}{{s_{0}( {{mIT}_{e} + {jT}_{e}} )} = {{\sum\limits_{n = {- L_{0}}}^{L_{0}}{{h_{0}( {{nIT}_{e} + {jT}_{e}} )}a_{m - n}\mspace{14mu}{for}\mspace{14mu} 0}} \leq j < I}} & (2)\end{matrix}$

The parameter L₀ is the half-length of the transmission filter which isspread over a duration of (2L₀+1)IT_(e). In the particular case of anNRZ transmission filter, L₀=0. As regards the transmitted signal s(t),this satisfies the equation s(t)=s₀(t)exp(j2Πf₀t) as it is equal to thesignal s₀(t) transposed to the frequency f₀. Under these conditions, theexpression s(mIT_(e)+jT_(e)) is, from (2):

$\begin{matrix}{\begin{matrix}{{s( {{mIT}_{e} + {jT}_{e}} )} = {\sum\limits_{n = {- L_{0}}}^{L_{0}}{{h_{0}( {{nIT}_{e} + {jT}_{e}} )}{\exp( {{j2\pi}\;{f_{0}( {{nI} + j} )}T_{\underset{\_}{e}}} )}}}} \\{a_{m - n}{\exp( {{j2\pi}\;{f_{0}( {m - n} )}{IT}_{e}} )}} \\{= {{\sum\limits_{n = {- L_{0}}}^{L_{0}}{{h_{F\; 0}( {{nIT}_{e} + {jT}_{e}} )}b_{m - n}\mspace{14mu}{such}\mspace{14mu}{that}\mspace{14mu} 0}} \leq j < I}}\end{matrix}{{where}\mspace{14mu}\begin{matrix}{{h_{F\; 0}( {iT}_{e} )} = {{h( {iT}_{e} )}{\exp( {{j2}\;\pi\; f_{0}{iT}_{e}} )}\mspace{14mu}{and}}} \\{b_{i} = {a_{i}{\exp( {{j2\pi}\; f_{0}{iIT}_{e}} )}}}\end{matrix}}} & (3)\end{matrix}$

Reception of the Signals by the Sensors

The transmitted signal s(t) (FIG. 3) passes through a propagationchannel before being received on an array made up of N antennas. Thepropagation channel may be modelled by P multipaths of angle ofincidence θ_(p), delay τ_(p) and amplitude ρ_(p) (1≦p≦P). At the outputof the antennas is the vector x(t), which corresponds to the sum of alinear mixture of P multipaths and noise, assumed to be white andGaussian. This vector of dimension N×1 is given by the followingexpression:

$\begin{matrix}\begin{matrix}{{x(t)} = {{\sum\limits_{p = 1}^{P}{\rho_{p}{a( \theta_{p} )}{s( {t - \tau_{p}} )}}} + {b(t)}}} \\{\;{= {{A\;{s(t)}} + {b(t)}}}}\end{matrix} & (4)\end{matrix}$where ρ_(p) is the amplitude of the pth path, b(t) is the noise vector,assumed to be Gaussian, a(θ) is the response of the array of sensors toa source with angle of incidence θ, and A=[a(θ₁) . . . a(θ_(p))] ands(t)=[s(t−τ₁) . . . s(t−τ_(p))]^(T). Noting that τ_(p)=r_(p)T+Δτ_(p)(where 0≦Δτ_(p)<T=IT_(e) and r_(p) is an integer), and insertingequation (3) into equation (4), the vector received by the antennas isgiven by:

$\begin{matrix}{{x( {{mIT}_{e} + {jT}_{e}} )} = {{\sum\limits_{p = 1}^{P}{\sum\limits_{\;{n = {- L_{0}}}}^{L_{0}}{\rho_{p}{a( \theta_{p} )}{h_{F\; 0}( {{nIT}_{e} + {jT}_{e} - {\Delta\tau}_{p}} )}{b_{m - n - r}}_{p}}}} + {b( {{mIT}_{e\;} + {jT}_{e}} )}}} & (5)\end{matrix}$

By making the change of variable according to u_(p)=n+r_(p), the vectorreceived by the antennas is given by:

$\begin{matrix}{{x( {{mIT}_{e} + {jT}_{e}} )} = {{\sum\limits_{p = 1}^{P}{\sum\limits_{{u_{p} = {r_{p} - L_{0}}}\;}^{r_{p} + L_{0}}\;{\rho_{p}{a( \theta_{p} )}{h_{F\; 0}( {{( {u_{p} - r_{p}} ){IT}_{e}} + {jT}_{e\;} - {\Delta\tau}_{p}} )}b_{m - u_{p}}}}} + {b( {{mIT}_{e} + {jT}_{e}} )}}} & (6)\end{matrix}$

Notating r_(min)=min{r_(p)} and r_(max)=max{r_(p)}, equation (6) may berewritten as follows:

$\begin{matrix}{{x( {{mIT}_{e} + {jT}_{e}} )} = {{\sum\limits_{p = 1}^{P}{\sum\limits_{u = {r_{\min} - L_{0}}}^{r_{\min} + L_{0}}{\rho_{p}{a( \theta_{p} )}{h_{F\; 0}( {{( {u - r_{p}} ){IT}_{e}} + {jT}_{e\;} - {\Delta\tau}_{p}} )}{{Ind}_{\lbrack{{{rp} - {L\; 0}},{{rp} + {L\; 0}}}\rbrack}(u)}b_{m - u}}}} + {b( {{mIT}_{e} + {jT}_{e}} )}}} & (7)\end{matrix}$where Ind_([r,q])(u) is the usual indicatrix function (Ind_([r,q])(u)=1for r≦u≦p and Ind_([r,q])(u)=0 otherwise) defined over the set ofintegers relating to the value in the binary set {0,1}, characterized byInd_([r,q])(u)=1 if u belongs to the interval [r,q] and Ind_([r,q])(u)=0otherwise. Thus, denoting the channel vector by v(t):

$\begin{matrix}{{v( {{uIT}_{e} + {jT}_{e}} )} = {\sum\limits_{p = 1}^{P}{\rho_{p}{a( \theta_{p} )}{h_{F\; 0}( {{( {u - r_{p}} ){IT}_{e}} + {jT}_{e\;} - {\Delta\tau}_{p}} )}{{Ind}_{\lbrack{{{rp} - {L\; 0}},{{rp} + {L\; 0}}}\rbrack}(u)}}}} & (8)\end{matrix}$where t=uIT_(e)+jT_(e) and equation (5) becomes:

$\begin{matrix}{{{x( {{mIT}_{e} + {jT}_{e}} )}{\sum\limits_{u = {r_{\min} - L_{0}}}^{r_{\min} + L_{0}}{{v( {{uIT}_{e} + {jT}_{e}} )}b_{m - u}}}} + {b( {{mIT}_{e} + {jT}_{e}} )}} & (9)\end{matrix}$

Inter-Symbol Interference

The observation vector x(t) coming from the antenna array att=mIT_(e)+jT_(e) involves, from equation (9), the symbol b_(m), but alsothe symbols b_(m-u) where u is a relative integer lying within the[r_(min)−L₀, r_(max)+L₀] interval, which phenomenon is more widely knownas ISI (inter-symbol interference). Let L_(c) be the number of symbolsparticipating in the ISI and let the interval of values taken by thelatter be limited. From equation (9), if the intersection of theintervals [r_(p)−L₀, r_(p)+L₀] is non-empty, thenL_(c)=|r_(max)−r_(min)|+2L₀+1. Consequently, when r_(max)=r_(min), thatis to say when all the multipaths are correlated, the lower bound ofL_(c) is reached and is given by L_(c)=2L₀+1. This case is alsomathematically expressed by

${{{\max\limits_{p}\{ \tau_{p} \}} - {\min\limits_{p}\{ \tau_{p} \}}}} < {T.}$On the other hand, if the intersection of said intervals is empty and ifall the intervals [r_(p)−L₀, r_(p)+L₀] are disjoint, thenL_(c)=P×(2L₀+1), which constitutes an upper bound for the set of valuesthat can be taken by L_(c). The latter situation correspondsspecifically to the case of multipaths that are all pairwisedecorrelated. This may also be mathematically expressed as ∀i≠j,|r_(i)−r_(j)|>2L₀, this condition being obtained whenever|τ_(i)−τ_(j)|>(2L₀+1)T. To summarize, the quantity L_(c) is in generalbounded as follows:2L ₀+1≦L _(c) ≦P×(2L ₀+1)  (10)

The equation expressing the vector received by the sensors can then berewritten in the following manner, but this time only the L_(c) symbolsb_(m-u) of interest appear:

$\begin{matrix}{{X( {{mIT}_{e} + {jT}_{e}} )} = {{\sum\limits_{l = 1}^{L_{0}}{{h( {{{n(I)}{IT}_{e}} + {jT}_{e}} )}b_{m - {n{(I)}}}}} + {b( {{mIT}_{e} + {jT}_{e}} )}}} & (11)\end{matrix}$where ∀1≦I≦L_(c) and r_(min)−L₀≦n(I)≦r_(min)+L₀ and where:

$\begin{matrix}{{h(t)} = {\sum\limits_{p = 1}^{P}{\rho_{p}{a( \theta_{p} )}{h_{F0}( {t - \tau_{p}} )}}}} & (12)\end{matrix}$

ICA Techniques

The process uses ICA techniques based on the following model, given byway of entirely nonlimiting illustration:

$\begin{matrix}\begin{matrix}{u_{k} = {{\sum\limits_{i = 1}^{\; L}{g_{i}s_{ik}}} + n_{k}}} \\{= {{Gs}_{k} + n_{k}}}\end{matrix} & (13)\end{matrix}$where u_(k) is a vector of dimension M×1 received at time k, s_(ik) isthe ith component of the signal s_(k) at time k, n is the noise vectorand G=[g₁ . . . g_(L)]. The objective of the ICA methods is to extractthe I=L components s_(ik) and to identify their signatures g_(i) (thevectorial response of source i through the observation u_(k)) on thebasis of the observation u_(k). The number I=L of components must notexceed the dimension M of the observation vector. The methods ofreferences [4], [5] and [15] use 2nd- and 4th-order statistics of theobservations u_(k). The first step uses 2nd-order statistics for theobservations u_(k) (these observations may be functions of the signalsreceived by the sensors) in order to obtain a new observation z_(k),such that:

$\begin{matrix}\begin{matrix}{z_{k} = {W_{1}u_{k}}} \\{= {{\sum\limits_{i = 1}^{\; L}{{\overset{˘}{g}}_{i}s_{ik}}} + {\overset{\sim}{n}}_{k}}} \\{= {{\overset{˘}{G}s_{k}} + {\overset{\sim}{n}}_{k}}}\end{matrix} & (14)\end{matrix}$where the signatures {hacek over (g)}_(i) (1≦i≦L) are orthogonal, {hacekover (G)}=[{hacek over (g)}₁ . . . {hacek over (g)}_(L)] ands_(k)=[s_(1k) . . . s_(Lk)]^(T). The second step consists in identifyingthe orthogonal base of the {hacek over (G)} values using 4th orderstatistics of the whitened observations z_(k). In this way, the signalss_(k) may be extracted by effecting:ŝ _(k) ={hacek over (G)} ^(#) z _(k) ={hacek over (G)} ^(#) W ₁ u_(k)  (15)where Ŝ_(k) is the estimate of the signals s_(k) and where ^(#) is thepseudo-inversion operator defined by {hacek over (G)}^(#)=({hacek over(G)}^(H){hacek over (G)})⁻¹{hacek over (G)}^(H).

The ICAR method [19] uses only 4th-order statistics to identify thematrix G=[g₁ . . . g_(k)] of signatures.

To summarize, the idea employed in the process according to theinvention is to construct a spatio-temporal observation, the mixedsources of which are symbol trains from the transmitter. Each symboltrain is for example the same symbol train but shifted by an integralnumber of symbol periods T.

Several ways of implementing the method will be described below, some ofwhich are explained by way of non-limiting illustration.

First Way of Implementing the Process.

FIG. 7 shows a first illustrative way of implementing the process inwhich the signal is received in baseband.

The method comprises a step I.1 of determining the symbol time T_(e),for example by applying a cyclic detection algorithm, such as thatdescribed for example in [1] [10].

The next step I.2 consists in interpolating the observations x(t) with Isamples per symbol, such that T=IT_(e).

Under these conditions where f₀=0 and b_(k)=a_(k), equation (11) for thevector becomes:

$\begin{matrix}{{x( {{mIT}_{e} + {jT}_{e}} )} = {{{\sum\limits_{l = 1}^{L_{0}}{{h( {{{n(I)}{IT}_{e}} + {jT}_{e}} )}a_{m - {n{(I)}}}}} + {{b( {{mIT}_{e} + {jT}_{e}} )}\mspace{14mu}{for}\mspace{14mu} 0}} \leq j < I}} & (16)\end{matrix}$

Since equation (16) is valid for 0≦j<I, the method constructs the nextspatio-temporal observation (step I3) from the observations x(kT_(e)):

$\begin{matrix}{\begin{matrix}{{z( {mIT}_{e} )} = \begin{bmatrix}\begin{matrix}\begin{matrix}{x( {mIT}_{e} )} \\{x( {{mIT}_{e} + T_{e}} )}\end{matrix} \\\vdots\end{matrix} \\{x( {{mIT}_{e} + {( {I - 1} )T_{e}}} )}\end{bmatrix}} \\{= {{\sum\limits_{l = 1}^{L_{c}}{{h_{z}( {n(I)} )}a_{m - {n{(I)}}}}} + {{b_{z}( {mIT}_{e} )}\mspace{14mu}{where}\mspace{14mu}{h_{z}(n)}}}} \\{= \begin{bmatrix}\begin{matrix}\begin{matrix}h_{n,0} \\h_{n,1}\end{matrix} \\\vdots\end{matrix} \\h_{n,{I - 1}}\end{bmatrix}}\end{matrix}\begin{matrix}{{{with}\mspace{14mu} h_{n,j}} = {{h( {{nIT}_{e} + {jT}_{e}} )}\mspace{14mu}{and}}} \\{{b_{z}( {mIT}_{e} )} = {\lbrack {{b( {mIT}_{e} )}^{T}\ldots\mspace{11mu}{b( {{mIT}_{e} + {( {I - 1} )T_{e}}} )}^{T}} \rbrack^{T}.}}\end{matrix}} & (17)\end{matrix}$

Since it is known that x(t) has the dimensions N×1, the vector z(t) hasthe dimensions NI×1.

h(k) is a vector whose nth component is the kth component of the filterthat linearly filters the symbol train {a_(m)} on the nth sensor. Thefilter for the vector coefficient h(k) depends both on the wave-shapingfilter and on the propagation channel.

To extract the L_(c) symbol trains {a_(m-i)} of interest (the number ofsymbols participating in the ISI), the method samples the receivedsignal with I=(2L₀+1), assuming that P≦N.

Since it is known that the NRZ filter satisfies 2L₀+1=1 and the Nyquistfilter 2L₀+1=3 for a roll-off of 0.25, the symbol trains may beextracted for these two wave-shaping filters when P≦NI and 3P≦NI,respectively.

Having determined the observation vector z(t), the process applies anICA-type method to estimate the L_(c) symbol trains {a_(m-i)} associatedwith the channel vectors ĥ_(z,j)=ĥ_(z)(k_(j)).

The jth output of the ICA method gives the symbol train {â_(m,j)}associated with the channel vector ĥ_(z,j). The estimated symbol trains{â_(m,j)} arrive in a different order from that of the {a_(m-i)} trainssatisfying:â _(m,j)=ρexp(jα _(i))a _(m-i) and ĥ_(z,j)=ĥ_(z)(i)  (18)

The symbol trains {â_(m,j)} are estimated with the same amplitudebecause the symbol trains {a_(m-i)} all have the same power, satisfyingthe equation:E[|a _(m-n(1))|² ]= . . . =E[|a _(m-n(Lc))|²].

The next step I.4 of the process has the objective of ordering the L_(c)outputs (â_(m,j,)ĥ_(z,j)) in the same order as the inputs (a_(m-i),h_(z)(i)) so as to obtain the channel vectors ĥ_(z,j)=ĥ_(z)(k_(j)). Todo this, the process intercorrelates pairwise the outputs â_(m,i) andâ_(m,j), calculating the following criterion c_(i,j)(k):

$\begin{matrix}{{c_{i,j}(k)} = \frac{E\lbrack {{\hat{a}}_{m,i}{\hat{a}}_{{m - k},j}^{*}} \rbrack}{\sqrt{{E\lbrack {{\hat{a}}_{m,i}{\hat{a}}_{m,i}^{*}} \rbrack}{E\lbrack {{\hat{a}}_{{m - k},j}{\hat{a}}_{{m - k},j}^{*}} \rbrack}}}} & (19)\end{matrix}$

When the function |c_(i,j)(k)| is a maximum in k=k_(max), the ith andjth outputs satisfy the equation: â_(m,i)=â_(m-k max,j). The algorithmfor classifying the outputs â_(m,n(1)) . . . â_(m,n(Lc)) is for examplecomposed of the following steps:

Step A.1: Determination of the output â_(m,imax) associated with thechannel vector of higher-modulus ĥ_(z,jmax).

Step A.2: For all the outputs â_(m-k,j) where j≠i_(max), determinationof the indices k=k_(j) maximizing the |c_(imax,j(k))| criterion. Fromthis is deduced, for each j, that â_(m,imax)=â_(m-kj,j). Since it isknown that c_(imax,j)(k_(j))=exp(jα_(imax)−jα_(j)) the jth output isreset to the same phase as the iith i_(max) output by takingâ_(m-kj)=c_(imax,j)(k_(j))â_(m,j). The channel vectors are also reset interms of phase by taking: {circumflex over(ĥ)}_(z)(k_(j))=ĥ_(z,j)c_(imax,j)(k_(j))*.

Step A.3: This step reorders the outputs â_(m-kj) and the channelvectors {circumflex over (ĥ)}_(z,j)=ĥ_(z)(k_(j)) in the increasing orderof the K_(j), since it is known that â_(m)=â_(m,imax) and that{circumflex over (ĥ)}_(z)(0)=ĥ_(z,imax).

After these three steps, the symbol trains {â_(m-k)} associated with thechannel vectors ĥ_(z)(k_(j)) are obtained. Since it is known that theestimated symbols satisfy the equation â_(m-k)=exp(jα_(imax))a_(m-k),the last step of the process consists in estimating this phase α_(imax).To do this, the constellation of symbols a_(k) is firstly identifiedamong a database consisting of the set of possible constellations. Thisdatabase consists of known constellations such as nPSK, n-QAM. Each timethat a new constellation is detected or becomes known, this is added tothe database.

FIG. 6 shows an example of an 8-QAM constellation when α_(imax)=0 andα_(imax)≠0. In this implementation example, the process then includesthe following steps:

The next step I.5 consists in determining the output phase associatedwith the channel vector of higher modulus. To identify the constellationand determine the phase, the process performs, for example, thefollowing steps:

Step I.5=Steps B.1, B.2 and B.3

Step°B.1: Estimation of the positions of the states of the constellation(red points in the figure) by seeking the maximum of the 2D histogrammeof the points M_(k)=(real(â_(k)), imag(â_(k))). For a constellationconsisting of M states, M pairs (û_(m),{circumflex over (v)}_(m)) for1≦m≦M are obtained.

Step°B.2: Determination of the type of constellation by comparing theposition of the states (û_(m),{circumflex over (v)}_(m)) of theconstellation of {â_(k)} symbols with a database comprising the set ofpossible constellations. The closest constellation is made up of thestates (u_(m),v_(m)) for 1≦m≦M.

Step B.3: Determination of the phase α_(imax) by minimizing in the senseof the least squares the following system of equations:û _(m)=cos(α_(imax))u _(m)−sin(α_(imax))v _(m) and {circumflex over (v)}_(m)=sin(α_(imax))u _(m)+cos(α_(imax))v _(m) for 1≦m≦M.

The process may include a step of estimating the propagation channelparameters of angle θ_(p) and delay τ_(p), of equation (8) by thealgorithm proposed in [8]. The step consists in extracting firstly thevectors h(nIT_(e)+jT_(e)) for 0≦j<I from the channel vectorsĥ_(z)(n_(j)) defined in equation (17). Followed by construction of thematrix H=[h(n(1)IT_(e)) . . . h(n(L_(c))IT_(e))] from equation (11) withthe h(nIT_(e)+jT_(e)) values in order to apply the parametric estimationmethod [8] for the multipaths: (θ_(p), τ_(p)) 1≦p≦p.

Second Way of Implementing the Process

FIGS. 8 and 9 show schematically another way of implementing theprocess, which may include two variants corresponding to thedecorrelated multipath case and to the groupwise correlated multipathcase, respectively.

Decorrelated Multipath Case.

The signal is received in baseband with {b_(k)}={a_(k)}.

The multipaths, the delays of which satisfy the relationship|τ_(j)−τ_(i)|>(2L₀+1)T, have the advantage of being decorrelated withone another, satisfying the equation: E[s(t−τ_(i))s(t−τ_(j))*]=0. Byexamining equation (4), it may therefore be seen that it is sufficientto apply an ICA type method when P≦N to the observation x(t) in order toobtain the signals s(t−τ_(p)) for each of the multibars. Afterestimating the signals for the various multipaths, the processdetermines their powers in order to keep the signal s(t−τ_(pmax)) of themultipath of higher amplitude ρ_(pmax). This main path is determinedusing the fact that the outputs of the ICA methods asymptoticallysatisfy:

$\begin{matrix}\begin{matrix}{{x(t)} = {\sum\limits_{p = 1}^{P}{\rho_{p}{a( \theta_{p} )}{s( {t - \tau_{p}} )}}}} \\{= {\sum\limits_{p = 1}^{P}{{\hat{a}}_{i}{{\hat{s}}_{i}(t)}\mspace{14mu}{with}}}} \\{{{{\hat{s}}_{i}(t)} = {\frac{s( {t - \tau_{p}} )}{\sqrt{\gamma_{p}}}\mspace{14mu}{and}}}\mspace{14mu}} \\{{\hat{a}}_{i} = {\sqrt{\gamma_{p}}\rho_{p}{a( \theta_{p} )}}}\end{matrix} & (20)\end{matrix}$where γ_(p)=ρ_(p) ²E[|s(t−τ_(p))|²]. Since the vectors a(θ_(p)) arenormed, satisfying the equation a(θ_(p))^(H)a(θ_(p))=N, the path ofmaximum amplitude will be associated with the i_(max) ^(th) output whereα_(imax)=â_(imax) ^(H)â_(imax) is a maximum. From equation (3), theoutput ŝ_(imax)(t)=s(t−τ_(pmax)) satisfies the equation:

$\begin{matrix}{{{\hat{s}}_{i\;\max}( {{mIT}_{e} + {jT}_{e}} )} = {\sum\limits_{n = {- L_{0}}}^{L_{0}}{{h_{F\; 0}( {{nIT}_{e} + {jT}_{e} - \tau_{p\;\max}} )}a_{m - n}}}} & (21)\end{matrix}$such that 0≦j<Iand it is possible to constitute the following observation vector:

$\begin{matrix}\begin{matrix}{{z( {mIT}_{e} )} = \begin{bmatrix}\begin{matrix}\begin{matrix}{{\hat{s}}_{i\;\max}( {mIT}_{e} )} \\{{\hat{s}}_{i\;\max}( {{mIT}_{e} + T_{e}} )}\end{matrix} \\\vdots\end{matrix} \\{{\hat{s}}_{i\;\max}( {{mIT}_{e} + {( {I - 1} )T_{e}}} )}\end{bmatrix}} \\{= {\sum\limits_{n = {- L_{0}}}^{L_{0}}{{h_{z}(n)}a_{m - n}}}} \\{{{where}\mspace{14mu}{h_{z}(n)}} = {\begin{bmatrix}\begin{matrix}\begin{matrix}h_{n,0} \\h_{n,1}\end{matrix} \\\vdots\end{matrix} \\h_{n,{I - 1}}\end{bmatrix}\mspace{14mu}{and}}}\end{matrix} & (22)\end{matrix}$where h_(n,j)=h_(F0)(nIT_(e)+jT_(e)−τ_(pmax)). According to the model ofequation (22), it is sufficient to apply an ICA method to theobservation z(mIT_(e)) in order to estimate the 2L₀+1 symbol trains{a_(m-n)} with −L₀≦n≦L₀. To extract the angles of incidence θ_(p) of thepropagation channel, it is sufficient from equation (20) to find, foreach signature â_(i) (1≦i≦P), the maximum of criterionc(θ)=|a(θ)^(H)â_(i)|². To extract the delays τ_(i)−τ₁ of the propagationchannel, it is sufficient from equation (20) to find, for each signalŝ_(i)(t) (1≦i≦P), the maximum of the c(τ)=|ŝ_(i)(t−τ)ŝ₁(t)*|² criterion.

To summarize, this variant comprises, for example, the following steps:

Step II.a.1: Determination of the symbol period T, applying a cyclicdetection algorithm as in [1] [10].

Step II.a.2: Sampling of the observations x(t) with I samples per symbolsuch that T=IT_(e).

Step II.a.3: Application of an ICA method to the observations x(t) inorder to obtain ŝ_(i)(t) and â_(i) for 1≦i≦P.

Step II.a.4: Determination of the output i=imax where α_(i)=â_(i)^(H)â_(i) is its maximum.

Step II.a.5: Formation of the observation vector z(t) of equation (22)from the signal ŝ_(imax)(t).

Step II.a.6: Application of an ICA method for estimating the symboltrains {a_(m-n)} where −L₀≦n≦L₀. From the symbol trains is chosen thatone which is associated with the higher-modulus vector h_(z)(n), namely{â_(m)}.

Step II.a.7: Determination of the phase α_(imax) of the outputassociated with the higher-modulus vector h_(z)(n) applying steps B.1,B.2 et B.3.

Step II.a.8: Phase-resetting of the symbol train {â_(m)} by taking{circumflex over (â)}_(m)=â_(m)exp(−jα_(imax)). The symbol train{{circumflex over (â)}_(m)} constitutes the output of the demodulator ofthis subprocess.

Step II.a.9: Estimation of the propagation channel parameters, namelyangle θ_(p) and delay τ_(p), by maximizing, for 1≦i≦P the|a(θ)^(H)â_(i)|² and |ŝ_(i)(t−τ)ŝ₁(t)*|² criteria for the angles anddelays respectively.

General Case for any or Groupwise-Correlated Multipaths

In this variant, the diagram for which is given in FIG. 9, the processconsiders that some of the multipaths are correlated. Considering thatthe transmitter is received according to Q groups of correlatedmultipaths, the signal vector received by the sensors becomes, fromequation (4):

$\begin{matrix}\begin{matrix}{{x(t)} = {{\sum\limits_{q = 1}^{Q}\;{\sum\limits_{p = 1}^{P_{q}}{\rho_{p,q}{a( \theta_{p,q} )}{s( {t - \tau_{p,q}} )}}}} + {b(t)}}} \\{= {{\sum\limits_{q = 1}^{Q}{A_{q}\Omega_{q}{s( {t,{\underset{\_}{\tau}}_{q}} )}}} + {b(t)}}}\end{matrix} & (23)\end{matrix}$

Where A_(q)=[a(θ_(1,q)) . . . a(θ_(Pq,q))], Ω_(q)=diag([ρ_(1,q) . . .ρ_(Pq,q)]) and s(t, τ _(q))=[s(t−τ_(1,q)) . . . s(t−τ_(Pq,q))]^(T) withτ _(q)=[τ_(1,q) . . . τ_(Pq,q)]^(T). The following signals andsignatures are estimated as output of the separator by applying an ICAmethod:

$\begin{matrix}\begin{matrix}{\hat{A} = \lbrack {{\hat{a}}_{1}\mspace{11mu}\ldots\mspace{11mu}{\hat{a}}_{{PQ},Q}} \rbrack} \\{= {\lbrack {A_{1}U_{1}\mspace{11mu}\ldots\mspace{11mu} A_{Q}U_{Q}} \rbrack\Pi\mspace{14mu}{and}}} \\{{\hat{s}(t)} = {\Pi\begin{bmatrix}{V_{1\;}{s( {t,{\underset{\_}{\tau}}_{1}} )}} \\\vdots \\{V_{Q}{s( {t,{\underset{\_}{\tau}}_{Q}} )}}\end{bmatrix}}} \\{= \begin{bmatrix}{{\hat{s}}_{1}(t)} \\\vdots \\{{\hat{s}}_{P_{Q}x_{Q}}(t)}\end{bmatrix}}\end{matrix} & (24)\end{matrix}$where Π is a permutation matrix, U_(q)V_(q)=Ω_(q) and V_(q)E[s(t,τ_(q))s(t,τ _(q))^(H)]V_(q) ^(H)=I_(Pq). Thus, the paths decorrelatedsuch that E[s(t−τ_(p,q))s(t−τ_(p′,q′))*]=0 are received on differentchannels ŝ_(i)(t) and ŝ_(j)(t). The correlated paths whereE[s(t−τ_(p,q))s(t−τ_(p′,q′))*]≠0 are mixed in the same channel ŝ_(i)(t)and are present on P_(Q) at the same time. In the 1st step of thissubprocess, we use this result to identify the Q group of correlatedmultipaths. Taking the outputs i and j of the separator, the twofollowing hypotheses may be tested:

$\begin{matrix}{H_{0}\text{:}\{ {\begin{matrix}{{{\hat{s}}_{i}(t)} = {b_{i}(t)}} \\{{{\hat{s}}_{j}(t)} = {b_{j}(t)}}\end{matrix}\mspace{14mu}{and}\mspace{14mu} H_{1}\text{:}\mspace{14mu}\{ \begin{matrix}{{{\hat{s}}_{i}(t)} = {{\alpha_{i}{s( {t - \tau_{p}} )}} + {b_{i}(t)}}} \\{{{\hat{s}}_{j}(t)} = {{\alpha_{j}{s( {t - \tau_{p}} )}} + {b_{j}(t)}}}\end{matrix} } } & (25)\end{matrix}$where E[b_(i)(t) b_(j)(t−τ)*]=0 whatever the value of τ. Thus for the H₀hypothesis, no multipaths exist common to the two output i and j, andfor the H₁ hypothesis there is at least one of them. The test consistsin determining whether the outputs ŝ_(i)(t) and ŝ_(j)(t−τ) arecorrelated for at least one of the τ values satisfying |τ|<τ_(max). Todo this, the Gardner test [3] is applied, which compares the followinglikelihood ratio with a threshold:

$\begin{matrix}\begin{matrix}{{V_{ij}(\tau)} = {{- 2}K\;{\ln( {1 - \frac{{{{\hat{r}}_{ij}(\tau)}}^{2}}{{{\hat{r}}_{ii}(0)}{{\hat{r}}_{jj}(0)}}} )}\mspace{14mu}{with}}} \\{{{\hat{r}}_{ij}(\tau)} = {\frac{1}{K}{\sum\limits_{k = 1}^{K}{{{\hat{s}}_{i}(t)}{{\hat{s}}_{j}( {t - \tau} )}^{*}\mspace{14mu}{or}}}}} \\ {{V_{ij}(\tau)} < \eta}\Rightarrow{{hypothesis}\mspace{14mu} H_{0}}  \\ {{{And}\mspace{14mu}{V_{ij}(\tau)}} \geq \eta}\Rightarrow{{hypothesis}\mspace{14mu} H_{1}} \end{matrix} & (26)\end{matrix}$

The threshold η is determined in [3] in relation to a chi square lawwith 2 degrees of freedom. The output associated with the 1st output arefirstly sought by starting the test by 2<j≦P_(Q)×Q and i=1. Next,removed from the list of outputs are all those associated with the 1stwhich will constitute the 1st group with q=1. The same series of testsis then restarted with the other outputs not correlated with the 1stoutput in order to constitute the 2nd group. This operation will becarried as far as the last group where, in the end, no output channelwill remain. After the sorting, what will finally be obtained are:Â _(q) =A _(q) U _(q) and ŝ _(q)(t)=V _(q) s(t,τ _(q)) for (1≦q≦Q)  (27)

The angles of incidence θ_(p,q) are determined from the Â_(q) values for(1≦q≦Q) applying the MUSIC [1] algorithm to the Â_(q)Â_(q) ^(H) matrix.The matrices A_(q) are deduced from these goniometry values. Since is itknown that x_(q)(t)=Â_(q)ŝ_(q)(t)=A_(q)Ω_(q)s(t, τ _(q)), s(t, τ _(q))is deduced therefrom to within a diagonal matrix by taking ŝ(t,τ_(q))=A_(q) ^(#)X_(q)(t). Since the elements of ŝ(t,τ _(q)) are composedof the signals ŝ(t−τ_(p,q)), the delays τ_(p,q)−τ_(1,1) are determinedby maximizing the c(τ)=|ŝ_(q,p)(t−τ)ŝ_(1,1)(t)*|² criteria whereŝ_(q,p)(t) is the p^(th) component of ŝ(t,τ _(q)).

Since it is known that E[ŝ_(q)(t)ŝ_(q)(t)^(H)]=I_(Pq), that A_(q)^(H)A_(q)=N I_(Pq) and that Â_(q)ŝ_(q)(t)=A_(q)Ω_(q)s(t,τ _(q)), it isdeduced therefrom that the group of multipaths associated with thelargest amplitudes Ω_(q) maximizes the following criterion:cri(q)=trace(Â_(q) ^(H)Â_(q)). From this is deduced the best outputassociated with Â_(qmax) and ŝ_(qmax)(t). Since from equation (3) thevector s(t, τ _(qmax)) satisfies equation:

$\begin{matrix}\begin{matrix}{{s( {{mlT}_{e} + {jT}_{e,{\underset{\_}{T}}_{q\;\max}}} )} = \begin{bmatrix}{s( {{mlT}_{e} + {jT}_{e} - \tau_{q\;\max\; 1}} )} \\\vdots \\{s( {{mlT}_{e} + {jT}_{e} - \tau_{q\;\max\; P_{q\;\max}}} )}\end{bmatrix}} \\{= {\sum\limits_{n = {- L_{0}}}^{L_{0}}{{h_{F\; 0}( {{nlT}_{e} + {jT}_{e,{\underset{\_}{T}q\;\max}}} )}a_{m - n}}}} \\{\begin{matrix}{0 \leq j < {1{\mspace{11mu}\;}{and}\mspace{14mu}{with}\mspace{14mu} h_{F\; 0}}} \\( {{nlT}_{e} + {jT}_{e,{\underset{\_}{T}q\;\max}}} )\end{matrix} = \begin{bmatrix}{h_{F\; 0}( {{nlT}_{e} + {jT}_{e} - \tau_{{q\;\max},1}} )} \\\vdots \\{h_{F\; 0}( {{nlT}_{e} + {jT}_{e} - \tau_{{q\;\max},P_{q\;\max}}} )}\end{bmatrix}}\end{matrix} & (28)\end{matrix}$forit is possible to constitute the following observation vector fromequation (27):

$\begin{matrix}\begin{matrix}{{z( {mlT}_{e} )} = \begin{bmatrix}{{\hat{s}}_{q\;\max}( {mlT}_{e} )} \\{{\hat{s}}_{q\;\max}( {{mlT}_{e} + T_{e}} )} \\\vdots \\{{\hat{s}}_{q\;\max}( {{mlT}_{e} + {( {I - 1} )T_{e}}} )}\end{bmatrix}} \\{= {\sum\limits_{n = {- L_{o}}}^{L_{0}}{{h_{z}(n)}a_{m - n}}}} \\{{{where}\mspace{14mu}{h_{z}(n)}} = \begin{bmatrix}h_{n,0} \\h_{n,1} \\\vdots \\h_{n,{l - 1}}\end{bmatrix}}\end{matrix} & (29)\end{matrix}$where h_(n,j)=V_(qmax)h_(F0)(nIT_(e)+jT_(e),τ _(qmax)). From the modelof equation (29), it is sufficient to apply an ICA method to theobservation z(mIT_(e)) in order to estimate the 2L₀+1 symbol trains{a_(m-n)} such that −L₀≦n≦L₀.

To summarize, this variant comprises the following steps:

Step II.b.1: Determination of the symbol period T by applying a cyclicdetection algorithm as in [1] and [10].

Step II.b.2: Sampling of the observations x(t) with I samples per symbolsuch that T=IT_(e).

Step II.b.3: Application of an ICA method to the observations x(t) inorder to obtain ŝ(t) and Â from equation (24).

Step II.b.4: Sorting of the outputs according to Q groups of correlatedmultipaths in order to obtain Â_(q) and ŝ_(q)(t) for (1≦q≦Q): to dothis, a correlation test for all the output pairs i and j with thetwo-hypothesis test of equation (26). Firstly the outputs associatedwith the 1st output will be sought by starting the test for 2<j≦P_(Q)×Qand i=1. Next, removed from the list of outputs are all those associatedwith the 1st that will constitute the 1st group with q=1. The sameseries of tests is repeated with the other outputs that are notcorrelated with the 1st output in order to constitute the 2nd group.This operation is continued to the last group where in the end no outputchannel will remain.

Step II.b.5: Determination of the better group of multipaths where(Â_(q) ^(H)Âq) is a maximum in q=qmax.

Step II.b.6: Constitution of the observation vector z(t) of equation(29) from the signal ŝ_(qmax)(t).

Step II.b.7: Application of an ICA method for estimating the symboltrains {a_(m-n)} where −L₀≦n≦L₀. From the symbol trains is chosen thatone which is associated with the higher-modulus vector h_(z)(i), namely{â_(m-i)}.

Step II.b.8: Determination of the phase α_(imax) of the outputassociated with the higher-modulus vector h_(z)(i) applying steps B.1,B.2 and B.3.

Step II.b.9: Phase-resetting of the symbol trains {â_(m)} by taking{circumflex over (â)}_(m)=â_(m)exp(−jα_(imax)). The symbol train{{circumflex over (â)}_(m)} constitutes the output of the demodulator ofthis subprocess.

Step II.b.10: Estimation of the propagation channel parameters, namelythe angle θ_(q,p) and the delay τ_(q,p). The angles of incidence θ_(q,p)are determined from the Â_(q) values for (1≦q≦Q) applying the MUSIC [1]algorithm to the matrix Â_(q)Â_(q) ^(H). The matrices A_(q) are deducedfrom these goniometry values in order to deduce therefrom an estimate ofs(t, τ _(q)) taking ŝ(t,τ _(q))=A_(q) ^(#)X_(q)(t). Since the elementsof the ŝ(t,τ _(q)) are composed of the signals s(t−τ_(q,p)), the delaysτ_(p,q)−τ_(1,1) are determined by maximizing thec(τ)=|ŝ_(q,p)(t−τ)ŝ_(1,1)(t)*|² critera where ŝ_(q,p)(t) is the p^(th)component of ŝ(t,τ _(q)).

Another way of Implementing the Process Estimation of the CarrierFrequency and Deduction of the {a_(m)} Symbols.

This technique consists in estimating the carrier frequency f₀ of thetransmitter or the complex z₀=exp(j2πf₀T_(e)) in order thereafter todeduce the symbols {a_(m)} from the symbols {b_(m)}, taking, fromequation (3):a _(m) =b _(m)exp(−j2πf ₀ mIT _(e))=b _(m) z ₀ ^(−mI)  (30)

This step is applied after step I.4 of reordering the symbols and thechannel vectors. From equations (3), (17), (7) and (8), the followingchannel vectors are used:

$\begin{matrix}{{{\hat{h}}_{z}(n)} = {{\begin{bmatrix}{z_{0}^{nl}{h( {nlT}_{e} )}} \\{z_{0}^{{nl} + 1}{h( {{nlT}_{e} + T_{e}} )}} \\\vdots \\{z_{0}^{{nl} + {({l - 1})}}{h( {{nlT}_{e} + {( {I - 1} )T_{e}}} )}}\end{bmatrix}\mspace{14mu}{for}\mspace{14mu} n} \in \Omega}} & (31)\end{matrix}$where Ω={Ind_([rp−L0, rp+L0])(n)=1 for a p such that 1≦p≦P}

Since it is known that Ω={n₁< . . . <n_(Lc)}, a grand vector b isobtained from the vectors ĥ_(z)(n), such that:

$\begin{matrix}{w = \begin{bmatrix}{{\hat{h}}_{z}( n_{1} )} \\{{\hat{h}}_{z}( n_{2} )} \\\vdots \\{{\hat{h}}_{z}( n_{K} )}\end{bmatrix}} & (32)\end{matrix}$

The search for f₀ consists in maximizing the following criterion:

$\begin{matrix}{{{{Carrier}\mspace{11mu}( f_{0} )} = {{w^{H}{c( {\exp( {{j2\pi}\; f_{0}T_{e}} )} )}}}^{2}}{{{where}\mspace{14mu} c( z_{0} )} = {\begin{bmatrix}{c( {n_{1},z_{0}} )} \\{c( {n_{2},z_{0}} )} \\\vdots \\{c( {n_{K},z_{0}} )}\end{bmatrix}\mspace{14mu}{and}}}\text{}{{{where}\mspace{14mu}{c( {n,z_{0}} )}} = \begin{bmatrix}z_{0}^{nl} \\z_{0}^{{nl}\; + \; 1} \\\vdots \\z_{0}^{{nl}\; + \;{({l\; + \; 1})}}\end{bmatrix}}} & (33)\end{matrix}$

The steps of the process suitable for the case of a transmitter with anon-zero frequency are the following:

Step III.a.1: Step I.1 to step I.4 described above in order to obtainthe symbol trains {{circumflex over (b)}_(m-k,)} associated with thechannel vectors ĥ_(z)(k_(j)).

Step III.a.2: Construction of the vector w of equation (32) from theĥ_(z)(k_(j)).

Step III.a.3: Maximization of the carrier (f₀) criterion of equation(33) in order to obtain f₀.

Step III.a.4: Application of equation (30) in order to deduce thesymbols {a_(m)} from the symbols {b_(m)}.

Step III.a.5: Step I.5 to step I.7 described above.

-   -   In the case of a transmitter with non-zero frequency and for        decorrelated multipaths, the steps are the following:

Step III.b.1: Step II.a.1 to Step II.a.4 described above in order toobtain the vector z(t) of equation (22).

Step III.b.2: Application of the ICA methods [4], [5], [15] and [19] inorder to estimate L_(c) symbol trains {{circumflex over (b)}_(m,j)}associated with the channel vectors ĥ_(z,j).

Step III.b.3: Reordering of the symbol trains {{circumflex over(b)}_(m,j)} and of the channel vectors ĥ_(z,j) applying steps A.1, A.2and A.3 in order to obtain the symbol trains {{circumflex over(b)}_(m-k,)} associated with the channel vectors ĥ_(z,j)=ĥ_(z)(k_(j)).

Step III.b.4: Construction of the vector w of equation (32) from theĥ_(z)(k_(j)).

Step III.b.5: Maximisation of the carrier (f₀) criterion of equation(33) in order to obtain f₀.

Step III.b.6: Application of equation (30) in order to deduce thesymbols {a_(m)} from the symbols {b_(m)}.

Step III.b.7: Choice of the symbol train associated with thehigher-modulus vector h_(z)(i), namely, {â_(m-i)}.

Step III.b.8: Step II.a.7 to step II.a.9 described above.

In the case of a transmitter with non-zero frequency and for correlatedmultipaths, the steps are for example the following:

Step III.c.1: Step II.b.1 to step II.b.6 No. 2.2 in order to obtain thevector z(t) of equation (29).

Step III.c.2: Application of ICA methods [4], [5], [15] and [19] inorder to estimate the L_(c) symbol trains {{circumflex over (b)}_(m,j)}associated with the channel vectors ĥ_(z,j).

Step III.c.3: Reordering of the symbol trains {{circumflex over(b)}_(m,j)} and of the channel vectors ĥ_(z,j) applying steps A.1, A.2and A.3 so as to obtain the symbol trains {{circumflex over (b)}_(m-k)_(j) } associated with the channel vectors ĥ_(z,j)=ĥ_(z)(k_(j)).

Step III.c.4: Construction of the vector w of equation (32) from theĥ_(z)(k_(j)).

Step III.c.5: Maximization of the carrier criterion (f₀) of equation(33) in order to obtain f₀.

Step III.c.6: Application of equation (30) for deducing the symbols{a_(m)} from the symbols {b_(m)}.

Step III.c.7: Choice among the symbol trains of that one which isassociated with the higher-modulus vector h_(z)(i), namely {â_(m-i)}.

Step III.c.8: Step II.b.8 to step II.b.10 described above.

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1. A process for the blind demodulation of a linear-waveform source ortransmitter in a system including one or more sources and an array ofsensors and a propagation channel, said process comprising steps of:determining symbol period T and taking samples at T_(e), such thatT=IT_(e), wherein I is an integer number and T_(e) is the samplingperiod; constructing a spatio-temporal observation vector z(t), themixed sources of which are symbol trains from the transmitter, fromobservations x(t_(k)) taken at times t_(k), where the time t_(k)corresponds to kT_(e) where k is an integer; applying an IndependentComponent Analysis (ICA)-type method to the observation vector z(t) inorder to estimate for a number of input symbol trains {a_(m-i)}corresponding to a number of symbols L_(c) participating in aninter-symbol interference, the input symbol trains {a_(m-i)}corresponding to observations, where m and i are positive integers, theestimate outputs {â_(m,i)} being associated with the channel vectorsĥ_(z,j)=ĥ_(z)(k_(j)), z corresponding to a sensor in the array and jcorresponding to a number of an estimate output; arranging the L_(c)outputs (â_(m,j), ĥ_(z,j)) in the same order as the inputs(a_(m-i),h_(z)(i)) so as to obtain the propagation channel vectorsĥ_(z,j)=ĥ_(z)(k_(j)); and determining a phase α_(imax) associated withthe outputs.
 2. The process as claimed in claim 1, further comprisingestimating propagation channel parameters in order to determine acarrier frequency so as to compensate for the symbol trains in order toobtain the symbol trains in baseband.
 3. The process as claimed in claim1, further comprising a step of estimating angle θ_(p) and delay τ_(p)parameters of the propagation channel, where p is a positive integer.